Given here is a regular hexagon, of side length **a**, the task is to find the area of the biggest triangle that can be inscribed within it.

**Examples:**

Input:a = 6Output:area = 46.7654Input:a = 8Output:area = 83.1384

**Approach**:

It is very clear that the biggest triangle that can be inscribed within the hexagon is an equilateral triangle.

In triangleACD,

following pythagorus theorem,

(a/2)^2 + (b/2)^2 = a^2

b^2/4 = 3a^2/4

So, b = a√3

Therefore, area of the triangle,A = √3(a√3)^2/4= 3√3a^2/4

Below is the implementation of the above approach:

## C++

`// C++ Program to find the biggest triangle ` `// which can be inscribed within the hexagon ` `#include <bits/stdc++.h> ` `using` `namespace` `std; ` ` ` `// Function to find the area ` `// of the triangle ` `float` `trianglearea(` `float` `a) ` `{ ` ` ` ` ` `// side cannot be negative ` ` ` `if` `(a < 0) ` ` ` `return` `-1; ` ` ` ` ` `// area of the triangle ` ` ` `float` `area = (3 * ` `sqrt` `(3) * ` `pow` `(a, 2)) / 4; ` ` ` ` ` `return` `area; ` `} ` ` ` `// Driver code ` `int` `main() ` `{ ` ` ` `float` `a = 6; ` ` ` `cout << trianglearea(a) << endl; ` ` ` ` ` `return` `0; ` `} ` |

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## Java

`// Java Program to find the biggest triangle ` `// which can be inscribed within the hexagon ` ` ` `import` `java.io.*; ` ` ` `class` `GFG { ` ` ` `// Function to find the area ` `// of the triangle ` `static` `double` `trianglearea(` `double` `a) ` `{ ` ` ` ` ` `// side cannot be negative ` ` ` `if` `(a < ` `0` `) ` ` ` `return` `-` `1` `; ` ` ` ` ` `// area of the triangle ` ` ` `double` `area = (` `3` `* Math.sqrt(` `3` `) * Math.pow(a, ` `2` `)) / ` `4` `; ` ` ` ` ` `return` `area; ` `} ` ` ` ` ` `public` `static` `void` `main (String[] args) { ` ` ` `double` `a = ` `6` `; ` ` ` `System.out.println (trianglearea(a)); ` ` ` ` ` `} ` `//This Code is contributed by Sachin.. ` ` ` `} ` |

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## Python3

`# Python3 Program to find the biggest triangle ` `# which can be inscribed within the hexagon ` `import` `math ` ` ` `# Function to find the area ` `# of the triangle ` `def` `trianglearea(a): ` ` ` ` ` `# side cannot be negative ` ` ` `if` `(a < ` `0` `): ` ` ` `return` `-` `1` `; ` ` ` ` ` `# area of the triangle ` ` ` `area ` `=` `(` `3` `*` `math.sqrt(` `3` `) ` `*` `math.` `pow` `(a, ` `2` `)) ` `/` `4` `; ` ` ` ` ` `return` `area; ` ` ` `# Driver code ` `a ` `=` `6` `; ` `print` `(trianglearea(a)) ` ` ` `# This code is contributed ` `# by Akanksha Rai ` |

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## C#

`// C# Program to find the biggest triangle ` `// which can be inscribed within the hexagon ` ` ` `using` `System; ` ` ` `class` `GFG { ` ` ` `// Function to find the area ` `// of the triangle ` `static` `double` `trianglearea(` `double` `a) ` `{ ` ` ` ` ` `// side cannot be negative ` ` ` `if` `(a < 0) ` ` ` `return` `-1; ` ` ` ` ` `// area of the triangle ` ` ` `double` `area = (3 * Math.Sqrt(3) * Math.Pow(a, 2)) / 4; ` ` ` ` ` `return` `Math.Round(area,4); ` `} ` ` ` ` ` `public` `static` `void` `Main () { ` ` ` `double` `a = 6; ` ` ` `Console.WriteLine(trianglearea(a)); ` ` ` ` ` `} ` ` ` `// This code is contributed by Ryuga ` ` ` `} ` |

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## PHP

`<?php ` `// PHP Program to find the biggest triangle ` `// which can be inscribed within the hexagon ` ` ` `// Function to find the area ` `// of the triangle ` `function` `trianglearea(` `$a` `) ` `{ ` ` ` ` ` `// side cannot be negative ` ` ` `if` `(` `$a` `< 0) ` ` ` `return` `-1; ` ` ` ` ` `// area of the triangle ` ` ` `$area` `= (3 * sqrt(3) * ` ` ` `pow(` `$a` `, 2)) / 4; ` ` ` ` ` `return` `$area` `; ` `} ` ` ` `// Driver code ` `$a` `= 6; ` `echo` `trianglearea(` `$a` `); ` ` ` `// This code is contributed ` `// by inder_verma ` `?> ` |

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**Output:**

46.7654

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